Description

Fundamentals of Physics

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

Fundqmcnt
The importance of achieving a clea-r understanding of engineedng fun_damentals should not be underestimated. As a desiEn anal.vst whJ can_nol wair o gel sErred wi|Jr FEA. ou mighr 6nd this chaprer o be a lilrleheagr on theory. If so, do not get bogged down: you can proceed tolater chapten and reler to this one as necessary. n fact, you will findmany references to specilic sections n this chapter throughout the restof the book. You should always ake the time to undentand the con-cepts being discussed. n the end, this chapter will likely become a vervpowerfirl engineering companion n lour analysis hallenges.
First Principles
Body Under Externol ooding
When performing engineering analpis, you are viftually alr^ra),s on-cemed with how a body will behave under extemal oading. Newton,s
28
Chopter 2t Fundomentats
Iaws, or the laws that will most generally govem this behaior, are listedbelow.. First Lau: Abody lrill remain at rest or will continue its straightline motion with constant velocity if there is no unbalancedforce acting on it.. Suund Lrtu:'['he acceleration of a body will be proportional tothe resultant of all forces acting on it and in the direction of theresultant.. Thi.rd. au: Acton and reaction forces between interacting bod-ies will be equal in magnitude, collinear, and opposite in direc-tion.The most important engineering equation arising from these laws fol-lows:8q.2.1 r=Nwhere -F s the resultant force v€ctor, rz is the mass of the body underconsideratron, and a is its acceleration vector.Because acceleration is the time deri\,?tive of rclocity (do/dt), ^ d, G =nE, constrtutes he kneu momtnturn\/ector of a body, the above equationcan also be wntten a5 ollovls,ttq. .z F=4i=cIn other words, Newon's second aw may also be interpreted as statingthat the time rate of a body's change of momenhrm will be propor-tional to the resultant force acting on it and in the salne direction.
Fig. 2.1. Genenlftee body
d agir8 ( a). Re su t a nt force ancl monents (b), Second
*q;; Q
a) (b) G)
Fittt Prin.iples
29
The most useful tool for understanding and implementing the loadsand constmints, or boLndary conditions hat govern a body s beha\ior, isthe Fee boq d.iagram.'fbe general free bod, diagram above (a) reprc-sents lhe body in space removed from its operating s,r'steltl. li exter-nally applied loads and reaction forces are represented with vectoft onthe body. If the body isir, equililirium, all these orce vectors mus add upco zero, both in magnitude and direction.In the most general sense, externally applied loading on a lhree-dimen-sional gid body cannot only alter iLs Enslation, but its rotation as well.Refeming to resultant forces and moments (b) anct the second la$'equivalent (c) in Fig. 2.1, the conesponding spatial equalions ofmotion for a rigid body follow:
Eqs. .3 lr = o
La=n
where IF and M are the force and mornent vector sums, respectively,of all externally applied loading, including reactions, and H is the d't?gr.,-Idr mommtum vector of the body. Both tM and H must be calcdatedabout the same point on the body.Iig. 2.1 sho\,'E ree body motion where this point corresponds to G thecenter of gravrty ofthe body. For constrained motion, it corresponds toq the fixed point about which the body rotales. In Eqs. 2.3, the timederi€tive of H is a complex quantif to deal rith mathematically, butsumce it to sal that H is a function of both the angular velocity andangular acceleration of the body. Its inertia component is not the massof the body but its mass Mnart of inaria rc ror (I), which is a 3 x 3matrix comprised of mass momenls ofineftia (Iii), ^nd mass protlucts af inzr-,ia (1i), derived with respect to the body coordinale a-xes. hese quanti-ties describe how the mass of a rigid body is distributed with respect tothe chosen a-..es. he general equations for these quantities follow:
Eqs. .1 r,,= I(l +f)dtu
t tj= )'tah
where ?, J, and & are any combination of the three coordinaLe a.\es cho-
30
Choprer 2t Fundo,mon o,lt
Fig. 2.2. Jniaxial sqdng and ,.,damper syden (a). Planar '-body notion (b).
Constraining the body to uniaxial motion and allowing for an extemalspring and -<lamper in the system, as shoun in fig 2'2(a), Eq 21expands to the following:Eq. 2.5 r,= ni+'t+k'with i denoting the spring stiftress, t t}Ie damPer coemci€nt' and Ir' at aod t the b;y's resultant apPlied force, Position, velocity, and accel_emtion along the x axis.Constraining he body to pla ar motion [see Fig 2'2(b)], Eqs 23sim-pliry to fie fouo$,ing xPression.Eq. 2'6 >F = tucluo=toaIn Ore above equahon, ac is the vectonal acceleration of the center of€raviw (c.s.) oi tfr. Uoay. EMc is rhe sum of all momenis about' hei ro.'ooini. l. i *t. mass moment oI inertia ot the body about an axis ,t.-if , tft phne of motion through the ( g. and d is dre bodv'sangular accelerauon.Barring dyramic anal)ses, FEA will always deal with bodies in equilib'rium. 6y deEnition, a body in such a state must have zero acceleration'so that Ore result of all extemally appljed forces must be zero This tyPeof anatsis is called statia Although *fs condition sounds very limiting,
(b)
V{E-m--^-^

Search

Similar documents

Tags

Related Search

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks